Chapter 8.5, Problem 59E

To determine

To find: the number of rabbits after one year by using the given formula.

Answer to Problem 59E

  $144$ .

Explanation of Solution

Given:

The given table shows that number of rabbits:

Month	1	2	3	4	5	6
Pairs at start of a month	1	1	2	3	5	8

Calculation:

Let us consider the given formula:

Recursive: $a_1=1,a_2=1,a_n=a_{n-2}+a_{n-1}$ .

Explicit: $f_n=\frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^n-\frac{1}{\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^n,n\ge 1$ .

Now by recursive rule,

  $\begin{array}{l}{a}_n=a_{n-2}+a_{n-1}\\ 1,1,2,3,5,8\\ \because a_3=a_2+a_1=1+1=2\\ \therefore a_7=a_6+a_5=8+5=13\\ \therefore a_8=a_7+a_6=13+8=21\\ \therefore a_9=a_8+a_7=21+13=34\\ \therefore a_7=a_9+a_8=34+21=55\\ \therefore a_7=a_{10}+a_9=55+34=89\\ \therefore a_{12}=a_{11}+a_{10}=89+55=144\end{array}$

Now, by explicit rule,

  $f_{12}=\frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^{12}-\frac{1}{\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^{12}=144$

Hence, by both rules, the number of rabbits after one year is $144$ .